Logic, Epistemology, and the Unity of Science
Reviewed by: Dale Jacquette (University of Bern)
Mark van Atten in this author-edited volume brings together eleven previously published or at time of writing about to independently appear essays in the history of the phenomenology of mathematics. Kurt Gödel’s relation to the work of G.W. Leibniz, Edmund Husserl and L.E.J. Brouwer makes Gödel’s philosophy as influenced by these thinkers the connecting theme of van Atten’s studies. Van Atten in turn seems to be strongly influenced in his reading of Gödel’s involvement with the limits of logic clearing the way for a phenomenology of logical-mathematical reasoning by Hao Wang’s frequently cited interviews with and commentary on Gödel’s philosophy of logic and mathematics.
Gödel in his 1931 ‘Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme. Pt. 1’ proves that there are arithmetical truths that cannot be nontrivially deduced or formally algorithmically verified by logically sound decision method, except from syntactically inconsistent assumptions by which classically any proposition and its negation are validly deduced. Having rigorously demonstrated that there are unprovable arithmetical truths, Gödel cancels logicism from the stark Fregean choices of pure logicism versus some form of psychologism. He meticulously constructs an unprovable undecidable sentence of arithmetic with implied inherent plans for designing as many counterexamples as desired. All that is strictly needed is one steady counterexample, although Gödel’s method suggests an unlimited structurally isomorphic plurality.
Gödel in 1931 thereby explodes Fregean-Russellian pure logicism in philosophy of arithmetic. The result leaves him to consider what psychological, phenomenological, intuitive or intuitionistic alternative might hold the best promise of shoring up the gap left in the foundations of mathematics by the deductive incompleteness of infinitary first-order arithmetic with addition and multiplication functions together with an identity relation. If objective mind-independent pure logical form cannot be the answer, then the mind is surely somehow involved. Mind ‘sees’ that the Gödel sentence implying its own deductive unprovability must be true, at least if first-order arithmetic is to remain syntactically consistent. Gödel sentences are guaranteed by indirectly self-referential construction to fail in deduction and decision if they are true. Mind judges that the Gödel sentence is true if arithmetic is to be contradiction-free, although the sentence is so constructed as to be true only if it is deductively unprovable from logically consistent assumptions. Pure logicism’s loss is phenomenology’s gain in Gödel’s evolving philosophy of mathematics, on the Wang interpretation that van Atten favors.
Recalled interviews with Gödel indicate that his turn toward phenomenology especially in the Princeton years after fleeing Vienna around the time of the Nazi Aschluß was not merely coincidental, like a medievalist with a side-interest in Jean-Paul Sartre. Setting that nonstarter aside, what is not answered, which is understandable given scanty equivocal historical documentation, and less satisfyingly unaddressed on philosophical grounds in van Atten or for that matter Wang is whether Gödel turns to phenomenology after the 1931 limiting metalogical proofs, or whether Gödel’s always latent phenomenological tendencies might have motivated and philosophically inclined him toward the discovery of the formal deductive incompleteness and sound algorithmic undecidability of infinitary first-order arithmetic. We underestimate Gödel one way or the other if we cannot imagine either of these interpretations being true of his intellectual depth and development. If Gödel gravitates especially toward the thinkers van Atten highlights in his essay-chapter investigations of each in historical turn, from Leibniz in the seventeenth century to Husserl and Brouwer among his closer contemporaries, then Gödel like other philosophers is presumably seeking out ideological antipodes and fellow-travelers.
Gödel-1 turns toward phenomenology and intuitionism after 1931, almost out of desperation and surprise. It as though the incompleteness proofs drive Gödel-1 unexpectedly away from pure logic and deductively valid mechanical syntax manipulation, once the discovery is made. Gödel-2 was always at heart a phenomenologist and intuitionist. He is impressed as were some members of the unofficially named Vienna Circle after Albert Einstein’s success in emphasizing the observer’s role in relativity physics when judging the position, speed and like factors of objects moving in spacetime. The application to logic may prove that the reasoning like the observing subject in physics needs to be included in the determination of logical truth, that there is no truth without thought, along with many other theoretically juicy suggestions. The choice of historical-philosophical interpretations of Gödel as Gödel-1 or Gödel-2 is arguably a if not the fundamental problem in understanding Gödel’s complex relationship with his discoveries in metamathematical logic and sustained interest in psychology, phenomenology and intuitionism. Qualifying my general admiration for van Atten’s accomplishment in this book is therefore a touch of disappointment that the essays do not address or even acknowledge this essential interpretive challenge.
Gödel after ‘Unentscheidbare Sätze’ concludes that the incompleteness of first-order arithmetic implies that minds are not mere syntax-processing machines like logically consistent formal symbolic deductive logical systems and mechanical decision methods. Van Atten does not take up the topic here, but says, p. 129: ‘We will leave a discussion of Gödel’s efforts on the question of minds and machines for another time’. There is a footnote (83) attached at the bottom of the page that mentions an in-progress essay with Leon Horsten and Rudy Rucker titled ‘Evolving a Mind’. This must be an essential piece of the puzzle in trying to reconstruct Gödel’s intellectual involvement with psychology, phenomenology and logical-mathematical intuitionism.
Van Atten seems to prefer Gödel-1, although he does not thematize in this way the history and philosophical dimensions of Gödel’s reception of Leibniz, Husserl and Brouwer. Nor does he recognize or try to argue the matter one way or another. He does not juxtapose the interpretations labeled here as Gödel-1 and Gödel-2 that could be recognized under any terminology. I find this a disappointing omission in the essays van Atten brings together in the book under review. It is one of the things that intrigues me most about the relation of Gödel’s metamathematics to his involvement with phenomenology and intuitionism, and I do not come away from van Atten’s discussions with a sense of how these things stand in Gödel’s thought.
There is surprisingly little said about Gödel’s proof at all in van Atten’s chapters, which as the book progresses becomes increasingly the unmentioned fabled elephant in the room. If van Atten has an opinion about the priorities of logical proof and intuition in Gödel’s thought, it would have been invaluable to have had his arguments and preferred interpretive analyses of this aspect of Gödel’s philosophy made explicit, the question raised even if only considered and deliberately unanswered. Gödel undoubtedly interested himself in phenomenology and intuitionism, as he did with respect to religious and mystic traditions, reflected in his personal library shelves inventoried at his death as reported by van Atten. The irrepressible historical-philosophical biographical question is which came first in Gödel’s lifework, the chicken of phenomenological and intuitionistic proclivities, or the deductive incompleteness egg of purely logically uncomprehended logical truth?
Working forward from Leibniz as the first important figure for Gödel in van Atten’s exposition, there seems to be an explanatory misconnection. Leibniz’s La Monadologie (1714) hypothesizes a God-chosen universal relation of interconnections among windowless monads that cannot bring about any changes in one another’s intrinsic natures or individual essences. Van Atten characterizes the parts of Leibniz’s metaphysics he regards as significant for Gödel in set theoretical language. The relation in Leibniz however is not naturally characterized in set theory, but more a matter of mereology, of part-whole or inferential connections among the truths of property instantiations by which each distinct monad is defined. If there are set theoretical commitments in Leibniz’s Monadology, they can only emerge after heavy interpretive overlay, given that set theory in anything like the modern sense does not appear in the history of mathematics as van Atten knows better than most until the mid-nineteenth century.
Van Atten relies heavily on Leibnizian references to the ‘reflection’ and ‘reflectiveness’ of each monad in every other monad distributed throughout the universe, but he does not explain what he takes Leibniz to mean by reflectiveness. Monad inter-reflectiveness in Leibniz is arguably better regarded as a purely abstract inferential network. Every monad is logically inferentially connected with every other monad if each monad’s interrelational properties is considered as its haecceity or uniquely individuating essence consisting of all its identifying conditions. Take any part of the universe and its relations like Leibniz’s contemporary Isaac Newton’s universal gravitation in which every physical object touches, attracts or repels but anyway affects every other object no matter how distant or with how weak and practically negligible a force. Leibniz’s monadology makes it possible in principle analogously to deduce from any object’s haecceity the haecceity of every other object. Information about all mutually causally untouchable interactively free unchangable Leibnizian monads is already fully contained in the information load of any and every monad. The role of set theory in understanding what Leibniz seems to mean beyond the summary just sketched seems negligible in identifying what Gödel might have found interesting in the inferential network of information about individual haecceities of all monads in Leibniz’s God-willed universe, the interlinkages of truths or truth-makers united together holistically in an unimaginably vast system of deductive implicational connections. That Gödel is highly interested in Leibniz and in set theory is not in dispute. The question is whether van Atten rightly interprets Gödel’s reasons for curiousity about Leibniz as plausibly explainable in set theoretical terms.
I admit to being confused by some aspects of van Atten’s recounting of Gödel’s interest in Husserl’s phenomenology. Gödel seems to have studied Husserl’s Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie (1913) with some care, as he did Husserl’s Logische Untersuchungen (1900/1913) and popularly more accessible 1929 Paris Sorbonne Lectures published in translation as Cartesianische Meditationen und Pariser Vorträge (1931). An example of the difficulty I had of following van Atten’s thread appears on p. 45. There van Atten begins with a substantial quotation from Wang’s (1996) A Logical Journey entries in his personal notebooks from 8.7.13-14. The passage is insightful, but attribution of the view expressed is first extended without further ado from Wang to Gödel. This is reasonable if in fact Wang is recalling the details of his conversations with Gödel. Remarkably, if I parse these passages correctly, the same proposition is then ascribed to Leibniz, again without special preparation or segue that I could uncover after several attempts squinting in the light of my desk lamp.
Van Atten in this instance writes: ‘The approach is “theological” [to adopt Wang’s language in the quoted text] because in the monadological setting, it is a central monad or God who creates a universe of objects.’ This may be true as far as it goes about Leibniz, but it is unclear from van Atten’s surround discussion whether Wang is exactly quoting Gödel and whether either Wang or Gödel would have had Leibniz’s monadology in mind in mentioning ‘monads’ and ‘the closeness aspect to what lies within the monad and in between the monads’. Leibniz is not the only thinker to invoke monads, and nothing prevents Wang or Gödel from picking up a useful terminology and turning it to their own very different non-Leibnizian ends. These are relations that for whatever reasons of lacunae in my education I anyway do not recognize as belonging to the Leibniz with whom I am familiar in the relatively late work Monadology, relatively early Discours de métaphysique (1686), Nouveaux essais sur l’entendement humain (1764), or others of his major writings on speculative metaphysics and scientific method. Perhaps the associations with Leibniz are obvious upon delving more deeply into Wang and Gödel as van Atten has, but things did not piece themselves together in my own efforts to connect the dots as van Atten presents them in his historical-philosophical narrative.
The linkage between Gödel and Husserl and Brouwer is more easily understood than Gödel’s fascination with Leibniz. Beyond its integrated metaphysics of logical interconnections and every logician’s taking Leibniz’s projection of a Characteristica universalis as an ideal for symbolic logic’s formal aspirations, as well as a German ancestor of logic, mathematics and so much more, it is not obvious at first what might have interested Gödel in Leibniz’s philosophy. There is a potential tension in van Atten’s efforts to subjoin Gödel’s interest in and affinity with Leibniz understood as set theoretical relations among representations of any monad’s properties with every other’s, and Husserl’s rejection of a specifically representational phenomenology. Leibnizian ‘reflection’ and ‘reflectiveness’ among monads understood as van Atten seems to interpret it as some kind of representation of their respective contents is not immediately compatible with Husserl’s rejection of representation in the phenomenology of perception.
Husserl’s reasonable argument is that mind does not represent an external reality if the two cannot be compared with one another for accuracy or inaccuracy of depiction. Given that one thought content can only be compared with another, there is no meaningful judgment of accuracy or inaccurancy of representation, and hence no sense in speaking of representation. If Leibniz’s ‘reflection’ and ‘reflectiveness’ of monads in other monads is understood set theoretically and representationally as van Atten seems to encourage, then there is a sudden breakdown between Leibniz and Husserl that van Atten does not acknowledge. It could be that there is in truth a basic disagreement between Leibniz and Husserl on the mutual representation of contents among monads, but that Gödel did not know it or fixed, on more positive applications of Husserl’s phenomenology, knew something about the incongruity but did not care. Did Gödel come to conclude that Leibniz so interpreted was right to regard monads as interconnected by representational ‘reflections’, or was he at some point convinced by Husserl that thought content does not represent in anything like the way that the plastic and performance arts, languages and artifacts can purposefully reference objects and states of affairs? It would be useful to consider attempts to rectify or smooth over the apparent disharmony in explaining the influence of these two thinkers on Gödel. The problem only arises on the assumption that what Leibniz means by the mutual reflectiveness of monads is representational. The difficulty disappears if reflectiveness is not interpreted representationally as van Atten proposes. The question is raised reading van Atten’s book, but the problem is not recognized nor answer provided.
Van Atten’s studies of Gödel’s interest in and influence on his philosophy of mathematics especially by Leibniz, Husserl and Brouwer, in different ways, with different effects and influences, despite the focus above on the reviewer’s burden of grudging critique, are extraordinarily rich in exploring the book’s chosen topics. Many more pages should be devoted to van Atten’s important contributions in the collection to begin to do it justice. The reader is strongly recommended to take up this detailed examination of Gödel’s selective reading in logic-related branches of phenomenological philosophy, as much for the questions it provokes as its detailed authoritative analysis of historical-philosophical themes.